How do I connect these two vertices by extending them along its edge util they reach each other?
Edit: grilled by Mr Zak  ;)
You can extend the 2 edges so that they overlap: slide GG and C to extend. Then enable the TinyCAD addon, select all and right click &gt; TinyCAD &gt; XALL Intersect, it will create a vertex at the intersection. Remove the rest.

Or, as mentionned by the Great Robin Betts, don't slide, just select all and choose VTX Auto, it will intersect automatically, you'll just have to dissolve the remaining vertices with CtrlX: